


Maximum Likelihood

by Bitenomnom



Series: Mathematical Proof [2]
Category: Sherlock (TV)
Genre: Gen, Mathematics, One explanation for some of the facts, Sherlock is obviously not a fan of Bayesian statistics
Language: English
Status: Completed
Published: 2012-09-06
Updated: 2012-09-06
Packaged: 2017-11-13 16:17:29
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 540
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/505385
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>“It’s <i>one</i> explanation of <i>some</i> of the facts, John,” said Sherlock, because that was the sort of thing that Sherlock said all the bloody time. </p><p>“Yeah, all right,” said John, “only he’s got a hole through his skull, an electric drill covered in blood by his head, and a note about power tools from somebody <i>admitting to the murder</i> stapled to his nose.”</p>
            </blockquote>





	Maximum Likelihood

**Author's Note:**

> I've done two for today, because I wrote this one, but it's sort of short, so I then wrote another, and like both. So I'll be posting the other shortly. This one is just some nice general math + Sherlock fun; the other is very much in the Johnlock vein.
> 
> "Maximum Likelihood Estimation," referred to by the prof of this class (Applied Linear Models) as "The Backwards Method" seems like it'd be Sherlock's sort of thing. (Incidentally, my prof for this class happens to be British!)
> 
> As before, first the stuff from class that inspired the writing, as an explanation for the content of the writing, then the writing itself.

An equation-free summary of some of my notes:

**Maximum Likelihood Estimation**

Make some observations. Pretend you haven’t seen the data. Suppose you have some unknown observations which are independent random variables. You can write the joint probability distribution—a function of the parameters given that you know the observations. But you can’t call that a probability distribution (where you know the parameters, but not the observations). Instead, it’s a function of the parameters where you know the observations. This we call the _likelihood function_. The _maximum likelihood estimation_ (MLE) of the parameter(s) is the value that maximizes the likelihood function. In other words: What parameters make the probability of observing what you observed the highest? _This is not the same as asking what the most likely observation was given the parameters._

An estimator of a parameter is unbiased (has no tendency to overestimate or underestimate the parameter) if the expected value of the estimator is equal to the parameter.

***

            “It’s _one_ explanation of _some_ of the facts, John,” said Sherlock, because that was the sort of thing that Sherlock said all the bloody time.

            “Yeah, all right,” said John, “only he’s got a hole through his skull, an electric drill covered in blood by his head, and a note about power tools from somebody _admitting to the murder_ stapled to his nose.”

            “And a metallic bird sculpture in his sitting room.”

            “What does that have to do with anything?”

            “Just one of the variables, John.” Sherlock placed his hands behind his back and strutted around the body before crouching back down near the puddle of oxidizing blood around the deceased man’s head.

            “And I suppose the type of biscuits he had with his tea today are one of the variables, too?” Of course they were, because _everything_ was, but John had had just about enough of Sherlock’s smartarse mouth for today and his patience was wearing thin.

            “Don’t be ridiculous,” Sherlock leaned down for a better look through the bloke’s head. “He didn’t have biscuits with his tea today.”

            “Just tell me what you know,” said Lestrade from across the room. His tired tone indicated that he was every bit as exasperated as John. Maybe, John thought, they’d go out for a pint later. If Sherlock solved this case today, he’d be back to reorganizing John’s wardrobe and peeling the prime-number pages out of the encyclopedia (what volume was he at now? twelve?) by seven o’clock.

            “As you have just heard me discuss with John, he’s dead, he has a drill beside him, he’s got a piece of paper attached to him, he’s got a metallic bird sculpture in his sitting room, and he didn’t have biscuits with his tea today,” Sherlock said, “for a start.”

            “So?”

            “So given all that, it was probably a suicide.”

            Both John and Lestrade’s eyes snapped to Sherlock, their lips parted in disbelief and eyebrows creased in confusion. “What?”

            “He wants his neighbor in jail for giving him that horrendous sculpture. Most likely explanation for the given observations.”

            “Huh,” John glanced at Lestrade and raised his eyebrows, because this most definitely was _not_ a suicide, and anyone with eyes (except, apparently, Sherlock) would know it. “Let it never be said Sherlock is unbiased.”


End file.
